$ B = \left[\begin{array}{rrr}4 & 1 & 3 \\ -2 & -1 & 1\end{array}\right]$ $ E = \left[\begin{array}{r}4 \\ 1 \\ -1\end{array}\right]$ Is $ B E$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ B$ , have? How many rows does the second matrix, $ E$ , have? Since $ B$ has the same number of columns (3) as $ E$ has rows (3), $ B E$ is defined.